Theorems · Theorem · potential theory
AnalyticAt.harmonicAt_log_norm
∀ {f : ℂ → ℂ} {z : ℂ}, AnalyticAt ℂ f z → f z ≠ 0 → InnerProductSpace.HarmonicAt (fun x => Real.log ‖f x‖) zIf f : ℂ → ℂ is complex-analytic without zero, then log ‖f‖ is harmonic.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 290 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- Norm.normstatement and proof · cited by 5,413
- Nat.cast_oneproof · cited by 2,501
- Nat.cast_zeroproof · cited by 1,870
- Real.logstatement and proof · cited by 939
- AnalyticAtstatement and proof · cited by 321
- Complex.slitPlaneproof · cited by 113
- Complex.normSqproof · cited by 103
- InnerProductSpace.HarmonicAtstatement and proof · cited by 23
- Complex.norm_defproof · cited by 18
Cited by4
Results whose statement or proof uses this declaration.
- circleAverage_re_herglotzRieszKernel_mul_log₀proof · cited by 1
- AnalyticOnNhd.circleAverage_log_norm_of_ne_zeroproof · cited by 1
- circleAverage_log_norm_sub_const₀proof · cited by 1
- circleAverage_log_norm_sub_const₂proof · cited by 1